The "definition" of the normal ordering in CFT looks a bit vague to me.
I found the definition in terms of exponentiated functional derivative pretty opaque.
Also in this context it might help if someone can give a reference or if there is a short explanation to understand how the Operator Product Expansion is derived using products of normal ordered operators.
I don't see the conceptual framework in which these ideas fit together.
Some of the books I looked at gave a very disparate view as a collection of some complicated formulas.
Let me give a precise example of the kind of calculation that I am stuck with,
Refer to these lecture notes
I can understand equation 4.26 of this but not the next 4 equations that seem to follow from it leading to 4.28.
It would be helpful if someone can decrypt the calculation.
In light of the kinds of references that came in as responses, I think it would help if I make the problematic calculation a little more explicit.
This has to do with what are called "Vertex Operators" in CFT given as $:e^{ikX(z)}:$ where $::$ is the notation for normal ordering and $k$ is some scalar and $X$ is a conformally invariant free Bosonic field. Then I would like to understand the derivation of this equality,
(all expressions are understood to be valid under the Feynman Path Integral)
$:\partial X(z)\partial X(z)::e^{ikX(w)}: = -\frac{k^2\alpha ^2}{4}\frac{:e^{ikX(w)}:}{(z-w)^2}-ik\alpha\frac{:\partial X(z) e^{ikX(w)}:}{(z-w)}$
where we have $X(z)X(w) = -\frac{\alpha}{2}ln \vert z – w \vert$
and what would be the similar simplification of
$:e^{ikX(z)}::e^{ikX(w)}: = ?$
Some more elaboration on what about normal ordering I am concerned about.
The problem is that I can't these books give an honest definition of what it means to "normal order" operators in CFT. Like there is a very clean definition in rest of QFT whose relation to time-ordering is given by the Wick's Theorem. Here in CFT one is supposed to understand that while normal ordering a string of operators inserted at different points on the space-time one is subtracting away from the product every possible way in which one or more pairs of insertion points can coincide and produce a singularity
Like if A,B,C,D are 4 different Bosonic operators say inserted at 4 different space-time points. Then one would define normal ordering as,
$:ABCD: = ABCD – (AB):CD: – (AC):BD: – (AD):BC:-(BC):AD:-(BD):AC:$
$$-(CD):AB:-(AB)(CD)-(AC)(BD)-(AD)(BC)$$
where () denotes the correlation function of the operators.
Now the point is whether one is supposed to take the above kind of equations as being just well-motivated definition or is there is anything more fundamental from which it is derivable?
There is definitely an issue about defining difference of two divergent expressions here.
Best Answer
The best reference I know for this sort of calculations is the PhD thesis of Kris Thielemans. Chapter 2 is all about the calculations of operator product expansions in two-dimensional conformal field theory. In particular in Section 2.6.1 you will find the kind of calculation you are interested in explained in detail. The formalism is clean and not very difficult at all.
Added
Here's a sketch of the calculation of the operator product expansion between two vertex operators $V_k(z)$ and $V_\ell(w)$ where $$ V_k(z) = :e^{i k X(z)}:,$$ where $$ X(z) = x - i p \log z + i \sum_{n\neq 0} \tfrac1n a_n z^{-n},$$ with canonical commutation relations (CCRs) $[p,x]=1$ and $[a_n,a_m] = n \delta_{n+m,0}$.
The normal ordering prescription consists of writing the creation operators to the left of the annihilation operators: $$ V_k(z) = e^{\sum_{n>0} \frac{k}{n} a_{-n} z^n} e^{ikx} z^{kp} e^{-\sum_{n>0} \frac{k}{n} a_n z^{-n}}.$$
To compute the operator product expansion $V_k(z) V_\ell(w)$ one simply has to take the formal product of the above expressions for the vertex operators and commute the operators through using the CCRs in order to bring the creation operators to the left. The basic calculations are simple Weyl identities of the form $$ e^{-\frac{k}{n}a_n z^{-n}} e^{\frac{\ell}{n} a_{-n} w^n} = e^{\frac{\ell}{n} a_{-n} w^n} e^{-\frac{k}{n}a_n z^{-n}} e^{\frac{k\ell}{n} (\frac{w}{z})^n} $$ and a similar one for $x$ and $p$.
The resulting power series converges provided that the fields are radially ordered so that $|z|>|w|$, but the result can be analytically continued with either a pole at $z=w$ or a branch cut singularity depending on the values of $k$ and $\ell$.
The final step is to expand the $z$-dependent fields around $z=w$.